Question

Which group of measurements could be the side lengths of a right triangle?

Answer 1

In right angled triangles, according to Pythagoras' theorem, squared of the hypotenuse is equal to the sum of the squares of the other 2 sides of the triangle.
if the hypotenuse is H and other 2 sides are A and B, then equation is as follows;
H² = A² + B²

1. A - 57 in. B - 86 in.
H² = A² + B²
H² = 57² + 86²
H² = 3249 + 7396
H  = √10645
    = 103.17 in.
third side given is not 103 in therefore this is not a right angled triangle 

2. A - 47 in. B - 76 in.
H² = A² + B²
H² = 47² + 76²
     = 2209 + 5776
H   = √7985
H = 89.3 in.
third side given is 95 in. therefore this is not a right angled triangle.

3. A- 57 in. + B - 76 in.
H² = A² + B²
H² = 57² + 76²
     = 3249 + 5776
H = √9025
H = 95 in.
third side given is 95 in. therefore this is a right angled triangle.

4. A - 57 in. B - 76 in.
As calculated in the third equation with the same side lengths of A and B, hypotenuse should be 95 in. however the third side length given is 105 in. so this is not a right angle triangle